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 07-18-2012, 12:17 AM #1 centurion   Join Date: Apr 2011 Posts: 150 Game theory: Optimal Strategy. Need help with a system of equations This question is from "The Mathematics of Poker". I have three equations based off a 3x3 Rock, Paper, Scissors payout table where winning with scissors is given a +2 payout instead of +1. Equations for Payoffs of choosing rock, paper, and scissors for player one, in order: -b+c a-2c -a+2b Solution: a=2b=2c Can someone show me how the solution was reached? Ironically, I am interested in this for reasons outside of poker. It is just cool what all you can turn into math. Thanks for the help
07-18-2012, 12:36 PM   #2
journeyman

Join Date: Sep 2004
Posts: 360
Re: Game theory: Optimal Strategy. Need help with a system of equations

You need to find a Nash equilibrium. You can use a zero-sum game solver:
http://levine.sscnet.ucla.edu/games/zerosum.htm

Quote:
 It is possible to solve a zero-sum game using the simplex algorithm or any other algorithm that can solve a linear programming problem.
I've no experience with this but:

-b+c=a-2c (1)
a-2c=-a+2b (2)

(1) a=-b+3c
(2) a=b+c

So b+c=-b+3c => b=c
So a = 2b = 2c
Frequency: a is 0.5 and b is 0.25 and c is 0.25

I guess something like this.

 07-18-2012, 01:33 PM #3 journeyman   Join Date: Sep 2004 Posts: 360 Re: Game theory: Optimal Strategy. Need help with a system of equations I guess my approach is incorrect ... it's probably ok to do it this way in this case (just because the solution is ok ) but not suitable to solve other pay-out tables.
 07-18-2012, 08:38 PM #4 centurion   Join Date: Apr 2011 Posts: 150 Re: Game theory: Optimal Strategy. Need help with a system of equations When you did the first steps of: -b+c=a-2c (1) a-2c=-a+2b (2) did you set those equations equal to each other because that it is a given that this is the first step in how you are supposed to solve these systems of equations? Also in: b+c=-b+3c => b=c How did you get b=c? Substitution, I'm guessing. and how did you use b=c to find: a = 2b = 2c I am not sure why b and c have coefficients of 2.
07-19-2012, 03:53 AM   #5

Join Date: Apr 2011
Posts: 747
Re: Game theory: Optimal Strategy. Need help with a system of equations

Quote:
 Originally Posted by MorrowCosom b+c=-b+3c => b=c
this should say
b + c = -b + 3c

add b to each side, subtract c
2b = 2c

07-19-2012, 04:26 AM   #6
centurion

Join Date: Apr 2011
Posts: 150
Re: Game theory: Optimal Strategy. Need help with a system of equations

Quote:
 this should say b + c = -b + 3c add b to each side, subtract c 2b = 2c
Out of some dumb-assing around I discovered this.

As for other games, I know the answer is a=b=c for the standard weighted rock, paper, scissors game going by "The Mathematics of Poker", but I cannot get that.

I come up with this when I try to work it:
-b+c
a-c
-a+b

1) -b+c=a-c => a=-b+2c
2) a-c=-a+b => 2a=b+c =>a=1/2b+1/2c

-b+2c=1/2b+1/2c
-2b+4c=b+c
-3b=-3c => b=c
3b=3c => b=c
so,
a=3b=3c

the correct answer is a=b=c according to the book. The equation in the up-top example of 2b=2c was not simplified to b=c, so I am pretty sure not simplifying at the end is not the problem.

Where did I mess up?

Sorry, the Physics Forum is just not responding.

Thanks again

07-19-2012, 04:48 AM   #7
journeyman

Join Date: Sep 2004
Posts: 360
Re: Game theory: Optimal Strategy. Need help with a system of equations

Quote:
 Originally Posted by MorrowCosom 1) -b+c=a-c => a=-b+2c 2) a-c=-a+b => 2a=b+c =>a=1/2b+1/2c ... 3b=3c => b=c
According to this a=b=c

a=-b+2b=b or a=-c+2c=c
a=1/2b+1/2b=b or a=1/2c+1/2c=c

But keep in mind I'm pretty sure this method is not suitable to solve all pay-out tables. I don't know where I use a shortcut. Better use the solver in the link I provided instead.

07-19-2012, 02:35 PM   #8

Join Date: Apr 2011
Posts: 747
Re: Game theory: Optimal Strategy. Need help with a system of equations

Quote:
 Originally Posted by MorrowCosom 1) -b+c=a-c => a=-b+2c 2) a-c=-a+b => 2a=b+c =>a=1/2b+1/2c
Quote:
 Originally Posted by MorrowCosom 3b=3c => b=c so, a=3b=3c
Substituting b=c into either 1) or 2) shows a=b or a=c as needed

 07-19-2012, 08:00 PM #9 journeyman   Join Date: Sep 2004 Posts: 360 Re: Game theory: Optimal Strategy. Need help with a system of equations I think I missed something when a pay-out table is asymmetric. I guess for symmetrical pay-out tables (like OP gave up as examples) it's easy: a, b and c are the same for both players. => You can use my method: Just make the equations (1) and (2) and solve it. It's easy to understand, expressions need to be equal to eachother and zero because of the symmetry. Example: You can't win/lose playing %Rock while your opponent loses/wins with %Rock. And if you both are winning or losing you are not in a Nash equilibrium (you are both vulnerable to exploitation). If you have asymmetrical pay-out tables it's different: a, b and c are different for both players. => You can't simply use my method: Equations in (1) and (2) cannot be made. Last edited by cyberfish; 07-19-2012 at 08:21 PM.
 07-19-2012, 08:23 PM #10 centurion   Join Date: Apr 2011 Posts: 150 Re: Game theory: Optimal Strategy. Need help with a system of equations When I gave an example of the first rock paper scissor game where the scissors had a weight of two the answer was a=2b=2c Why were b and c not simplified to b=c in the final answer, so that it was a=b=c? If 2b=2c, b=c. But, In the ordinary rock paper scissors game I worked where the answer was a=b=c : I got a=3b=3c and OmahaDonk, said that my answer was correct, because I could plug b=c into either equation 1) -b+c=a-c => a=-b+2c 2) a-c=-a+b => 2a=b+c =>a=1/2b+1/2c which means that 3b=3c is simplified. What am I missing? I would have worked them both like Omaha, but then would have missed the first answer because I simplified. Man, I am missing something little here. Thanks for the patience from the two of you.
 07-19-2012, 08:31 PM #11 journeyman   Join Date: Sep 2004 Posts: 360 Re: Game theory: Optimal Strategy. Need help with a system of equations Just make the correct substitutions (reread my 7th post). If a=-b+2c and b=c => a=-b+2b=b or a=-c+2c=c => a=b=c If a=b+c and b=c => a=b+b=2b or a=c+c=2c => a=2b=2c Last edited by cyberfish; 07-19-2012 at 08:45 PM.

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